L(s) = 1 | − 2-s − 1.66·3-s + 4-s + 1.81·5-s + 1.66·6-s + 3.25·7-s − 8-s − 0.218·9-s − 1.81·10-s − 4.39·11-s − 1.66·12-s − 0.979·13-s − 3.25·14-s − 3.02·15-s + 16-s + 7.76·17-s + 0.218·18-s + 0.157·19-s + 1.81·20-s − 5.42·21-s + 4.39·22-s − 23-s + 1.66·24-s − 1.70·25-s + 0.979·26-s + 5.36·27-s + 3.25·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.962·3-s + 0.5·4-s + 0.811·5-s + 0.680·6-s + 1.23·7-s − 0.353·8-s − 0.0729·9-s − 0.573·10-s − 1.32·11-s − 0.481·12-s − 0.271·13-s − 0.869·14-s − 0.781·15-s + 0.250·16-s + 1.88·17-s + 0.0516·18-s + 0.0360·19-s + 0.405·20-s − 1.18·21-s + 0.937·22-s − 0.208·23-s + 0.340·24-s − 0.341·25-s + 0.192·26-s + 1.03·27-s + 0.615·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.098410519\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.098410519\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 + 4.39T + 11T^{2} \) |
| 13 | \( 1 + 0.979T + 13T^{2} \) |
| 17 | \( 1 - 7.76T + 17T^{2} \) |
| 19 | \( 1 - 0.157T + 19T^{2} \) |
| 29 | \( 1 + 4.73T + 29T^{2} \) |
| 31 | \( 1 + 0.703T + 31T^{2} \) |
| 37 | \( 1 - 2.06T + 37T^{2} \) |
| 41 | \( 1 + 9.75T + 41T^{2} \) |
| 43 | \( 1 + 2.90T + 43T^{2} \) |
| 47 | \( 1 - 0.804T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 1.01T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 3.65T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 + 6.46T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 15.9T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 7.40T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.005006528694983462075636365231, −7.57186859872405733408260288257, −6.66700584585494268882633932945, −5.65828382285218087988594226374, −5.45613827043635181004367601133, −4.91195608281835740836652438847, −3.52385395581621238207933707310, −2.44256029623816688506700263028, −1.69556202473483218694640493863, −0.65160046760779705041873330009,
0.65160046760779705041873330009, 1.69556202473483218694640493863, 2.44256029623816688506700263028, 3.52385395581621238207933707310, 4.91195608281835740836652438847, 5.45613827043635181004367601133, 5.65828382285218087988594226374, 6.66700584585494268882633932945, 7.57186859872405733408260288257, 8.005006528694983462075636365231